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ITERATION AND CONTRACTION MAPPINGS
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Accordingly algebraic apply approximation argument axioms Banach space bounded bounded linear operators calculation Cauchy centre chain chapter Chebyshev polynomials closed ball co-ordinates coefficients condition consider continuous functions convergence corresponding curve deﬁned deﬁnition of distance derivative differentiation dimensions dual space error Euclidean space example exceed expression fact ﬁgure ﬁnd ﬁnding ﬁnite ﬁrst ﬁt ﬁxed formula Fourier series function f functional analysis geometrical given gives graph Hilbert space idea inﬁnite input instance integral equation interval inverse involved iteration justiﬁed limit linear operator mapping mathematics matrix maximum value means method metric space minimum Newton—Raphson numerical analysis obtain operator norm origin output perpendicular plane possible problem proof properties prove quadratic question real number result satisﬁes scalar product sequence signiﬁcant solution speciﬁed sphere square subspace suppose supremum symbol tangent tends theorem theory triangle inequality vector space whole number